77 lines
3.7 KiB
Text
77 lines
3.7 KiB
Text
Free oscillations in one dimension
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59
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Two independent solutions of the linear differential equation (21.5) are
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cos wt and sin wt, and its general solution is therefore
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COS wt +C2 sin wt.
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(21.7)
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This expression can also be written
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x = a cos(wt + a).
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(21.8)
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Since cos(wt+a) = cos wt cos a - sin wt sin a, a comparison with (21.7)
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shows that the arbitrary constants a and a are related to C1 and C2 by
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tan a = - C2/C1.
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(21.9)
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Thus, near a position of stable equilibrium, a system executes harmonic
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oscillations. The coefficient a of the periodic factor in (21.8) is called the
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amplitude of the oscillations, and the argument of the cosine is their phase;
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a is the initial value of the phase, and evidently depends on the choice of
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the origin of time. The quantity w is called the angular frequency of the oscil-
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lations; in theoretical physics, however, it is usually called simply the fre-
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quency, and we shall use this name henceforward.
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The frequency is a fundamental characteristic of the oscillations, and is
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independent of the initial conditions of the motion. According to formula
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(21.6) it is entirely determined by the properties of the mechanical system
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itself. It should be emphasised, however, that this property of the frequency
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depends on the assumption that the oscillations are small, and ceases to hold
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in higher approximations. Mathematically, it depends on the fact that the
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potential energy is a quadratic function of the co-ordinate.
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The energy of a system executing small oscillations is E =
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= 1m(x2+w2x2) or, substituting (21.8),
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E =
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(21.10)
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It is proportional to the square of the amplitude.
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The time dependence of the co-ordinate of an oscillating system is often
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conveniently represented as the real part of a complex expression:
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x = re[A exp(iwt)],
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(21.11)
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where A is a complex constant; putting
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A = a exp(ix),
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(21.12)
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we return to the expression (21.8). The constant A is called the complex
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amplitude; its modulus is the ordinary amplitude, and its argument is the
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initial phase.
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The use of exponential factors is mathematically simpler than that of
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trigonometrical ones because they are unchanged in form by differentiation.
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t It therefore does not hold good if the function U(x) has at x = 0 a minimum of
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higher order, i.e. U ~ xn with n > 2; see §11, Problem 2(a).
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60
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Small Oscillations
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§21
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So long as all the operations concerned are linear (addition, multiplication
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by constants, differentiation, integration), we may omit the sign re through-
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out and take the real part of the final result.
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PROBLEMS
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PROBLEM 1. Express the amplitude and initial phase of the oscillations in terms of the
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initial co-ordinate xo and velocity vo.
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SOLUTION. a = (xx2+002/w2), tan a = -vo/wxo.
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PROBLEM 2. Find the ratio of frequencies w and w' of the oscillations of two diatomic
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molecules consisting of atoms of different isotopes, the masses of the atoms being M1, m2 and
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'M1', m2'.
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SOLUTION. Since the atoms of the isotopes interact in the same way, we have k = k'.
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The coefficients m in the kinetic energies of the molecules are their reduced masses. Accord-
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ing to (21.6) we therefore have
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PROBLEM 3. Find the frequency of oscillations of a particle of mass m which is free to
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move along a line and is attached to a spring whose other end is fixed at a point A (Fig. 22)
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at a distance l from the line. A force F is required to extend the spring to length l.
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A
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X
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FIG. 22
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SOLUTION. The potential energy of the spring is (to within higher-order terms) equal to
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the force F multiplied by the extension Sl of the spring. For x < l we have 81 = (12++2) -
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=
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x2/21, so that U = Fx2/21. Since the kinetic energy is 1mx2, we have = V(F/ml).
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PROBLEM 4. The same as Problem 3, but for a particle of mass m moving on a circle of
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radius r (Fig. 23).
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m
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&
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FIG. 23
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